Problem 25 - Algebra and Trigonometry

Question

A pair of points is graphed. (a) Plot the points in a coordinate plane. (b) Find the distance between them. (c) Find the mid-point of the segment that joins them. \((0,8),(6,16)\)

Step-by-Step Solution

Answer

Plot the points, distance is 10 units, and the midpoint is (3,12).

Step 1: Plot the Points

Identify the given points, which are \((0,8)\) and \((6,16)\).Plot these points on the coordinate plane. Start by plotting \((0,8)\) by moving 8 units up from the origin along the y-axis. Then, plot \((6,16)\) by going 6 units to the right along the x-axis and then 16 units up.

Step 2: Calculate the Distance Between Points

Use the distance formula to find the distance between the points \((x_1, y_1)\) and \((x_2, y_2)\):\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute the coordinates \((0,8)\) and \((6,16)\):\[d = \sqrt{(6 - 0)^2 + (16 - 8)^2} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10\]The distance between the two points is 10 units.

Step 3: Find the Midpoint

Use the midpoint formula to find the midpoint of the line segment joining the points \((x_1, y_1)\) and \((x_2, y_2)\):\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Substitute the coordinates \((0,8)\) and \((6,16)\):\[M = \left( \frac{0 + 6}{2}, \frac{8 + 16}{2} \right) = \left( \frac{6}{2}, \frac{24}{2} \right) = (3,12)\]The midpoint of the segment that joins them is \((3, 12)\).

Key Concepts

Midpoint Formula

The midpoint formula is a simple yet powerful tool in geometry that allows you to find the center point of a line segment. This midpoint is like a balancing point, situated exactly halfway between two endpoints. To find the midpoint, use the formula:\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]Here's what each part of the formula represents:- \(x_1\) and \(x_2\) are the x-coordinates of the two points.- \(y_1\) and \(y_2\) are the y-coordinates of the two points.The result \((x, y)\) gives you the exact center of the line segment. In our example, for the points \((0,8)\) and \((6,16)\), the midpoint calculated is \((3,12)\).
This means if you were to draw a point at \((3,12)\), it would lie perfectly centered between \((0,8)\) and \((6,16)\) on the coordinate plane.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface where you can plot points, lines, and curves. It consists of two number lines that intersect perpendicularly:- The horizontal line is called the x-axis.- The vertical line is called the y-axis.These axes divide the plane into four quadrants. Each point on the plane is expressed as an ordered pair \((x, y)\), where \(x\) and \(y\) denote the point’s position relative to the origin, which is the center point \((0, 0)\) where both axes meet.
In the exercise, points \((0,8)\) and \((6,16)\) are located in the upper right quadrant, which is the first quadrant where both \(x\) and \(y\) are positive. Understanding this plane is crucial since it forms the foundation for graphing mathematical relationships.

Graphing Points

Graphing points on a coordinate plane is an essential skill in mathematics that allows you to visually interpret data and functions. To graph a point, follow these steps:

• Identify the coordinates. Each point has an x-value and a y-value, such as \((0,8)\) or \((6,16)\).
• Start at the origin \((0,0)\), the central point on the plane.
• Move horizontally first: if the x-coordinate is positive, move right; if negative, move left.
• From there, move vertically based on the y-coordinate: if positive, move up; if negative, move down.

Once you plot the points, they help in visualizing the relationships and distances between them. In the given task, plotting \((0,8)\) involves moving 8 units up from the origin, while plotting \((6,16)\) requires moving 6 units right and 16 units up. Graphing allows you to see how points relate to each other spatially, setting the stage for finding distances and midpoints with formulas.